You'll be fine in trail runners. By July, the snow should be soft enough that you won't have to worry about kicking in steps. If you wear boots, they're going to be full of water after every creek crossing--trail runners will dry much faster.

I'm going north to south starting a week after you. Should be the trip of a lifetime!

My daughter and I were on the section between Red's Meadow and Silver Pass last year, June 28-July 1. This year's snowfall has been slightly heavier than last year's. The area around Silver Pass was somewhat snowy, but nothing major. My daughter got through the snow in pajama pants and Vans, believe it or not! You're going to get wet no matter what, both from stream crossings and, to a lesser extent, from snow on high passes. Just make sure to wear wool socks, not cotton.